7.2
Problems
215
¯ ˆ
Let us check the behaviour of the current density ˆ j
μ (x) = qψ(x)γ
μ ψ ˆ (x)
em
under the transformation (7.169). Recalling that in the standard representation iα 1 α 3 = Σ 2 , we find
T ˆ ˆ j
0 (x, t)T ˆ −1 = ˆ j
0 (x, −t)
em
em
T ˆ j ˆ (x, t)T ˆ −1 = qψ ˆ † (x, −t)Σ 2 α
∗ Σ 2 ψ ˆ (x, −t) = −j ˆ (x, −t). (7.171)
em
em
This is exactly how A
μ (x), and hence A ˆ μ (x), transforms, and hence the electromagnetic interaction − ˆ j
μ A ˆ μ is T-invariant. The same is true in the KG
em
case.
We may now proceed to look at some simple processes in scalar and spinor
electrodynamics, in the following two chapters.
Problems
ˆ
′ ˆ′
7.1 Verify that the Lagrangian L ˆ of (7.1) is invariant (i.e. L ˆ (φ ˆ 1 , φ 2 ) = L ˆ (φ ˆ
1 , φ 2 ))
ˆ
′ ˆ ′
under the transformation (7.2) of the fields (φ ˆ 1 , φ 2 ) → (φ ˆ
1 , φ 2 ).
μ
(a) Verify that, for N ˆ given by (7.23), the corresponding N ˆ φ of (7.14)
φ
reduces to the form (7.24); and that, with H ˆ given by (7.21),
[N ˆ φ , H ˆ ] = 0.
(b) Verify equation (7.27).
7.3 Show that
[φ ˆ (x 1 ), φ ˆ † (x 2 )] = 0
for (x 1 − x 2 )
2 < 0
[Hint : insert expression (7.16) for the φ ˆ ’s and use the commutation relations (7.18) to express the commutator as the difference of two integrals; in
the second integral, x 1 − x 2 can be transformed to −(x 1 − x 2 ) by a Lorentz
transformation – the time-ordering of space-like separated events is framedependent!].
7.4 Verify that varying ψ
† in the action principle with Lagrangian (7.34) gives
the Dirac equation.
7.5 Verify (7.44).
7.6 Verify equations (7.52) and (7.53).
7.7 Verify (7.62).
Problems
215
¯ ˆ
Let us check the behaviour of the current density ˆ j
μ (x) = qψ(x)γ
μ ψ ˆ (x)
em
under the transformation (7.169). Recalling that in the standard representation iα 1 α 3 = Σ 2 , we find
T ˆ ˆ j
0 (x, t)T ˆ −1 = ˆ j
0 (x, −t)
em
em
T ˆ j ˆ (x, t)T ˆ −1 = qψ ˆ † (x, −t)Σ 2 α
∗ Σ 2 ψ ˆ (x, −t) = −j ˆ (x, −t). (7.171)
em
em
This is exactly how A
μ (x), and hence A ˆ μ (x), transforms, and hence the electromagnetic interaction − ˆ j
μ A ˆ μ is T-invariant. The same is true in the KG
em
case.
We may now proceed to look at some simple processes in scalar and spinor
electrodynamics, in the following two chapters.
Problems
ˆ
′ ˆ′
7.1 Verify that the Lagrangian L ˆ of (7.1) is invariant (i.e. L ˆ (φ ˆ 1 , φ 2 ) = L ˆ (φ ˆ
1 , φ 2 ))
ˆ
′ ˆ ′
under the transformation (7.2) of the fields (φ ˆ 1 , φ 2 ) → (φ ˆ
1 , φ 2 ).
μ
(a) Verify that, for N ˆ given by (7.23), the corresponding N ˆ φ of (7.14)
φ
reduces to the form (7.24); and that, with H ˆ given by (7.21),
[N ˆ φ , H ˆ ] = 0.
(b) Verify equation (7.27).
7.3 Show that
[φ ˆ (x 1 ), φ ˆ † (x 2 )] = 0
for (x 1 − x 2 )
2 < 0
[Hint : insert expression (7.16) for the φ ˆ ’s and use the commutation relations (7.18) to express the commutator as the difference of two integrals; in
the second integral, x 1 − x 2 can be transformed to −(x 1 − x 2 ) by a Lorentz
transformation – the time-ordering of space-like separated events is framedependent!].
7.4 Verify that varying ψ
† in the action principle with Lagrangian (7.34) gives
the Dirac equation.
7.5 Verify (7.44).
7.6 Verify equations (7.52) and (7.53).
7.7 Verify (7.62).